Test examples

The effectiveness of the dynamic grid was tested systematically using two series of adaptive meshes progressively refined with an increasing number of elements and nodes. The MG was coded on a standard PC, Intel(R) Core™2 i7, CPU870 at 2.93 GHz, running on Windows 7 with Intel FORTRAN VS2010 on XP mode. The domain employed is a 150 × 100 rectangle whose boundary is decomposed uniformly into line segments of length 10 units. The first node spacing function has been given by Equation 3.6, which specifies a higher-element density around a circle and an ellipse, as shown in Figure 3.50.

As for the second node spacing function, it requires elements to be concentrated on the perimeter of five intersecting circles, as shown in Figure 3.51. The mathematical expressions for this node spacing function are given by

ρ ρ

For the series of five meshes corresponding to the first node spacing function, ρmax = 20 and the minimum node spacing ρmin varies from 0.1 to 0.005, which represents a variation

Figure 3.50 FE mesh for the first nodal spacing function, Nn = 75,740, Ne = 151,428.

from 200 to 4000 in the linear size ratio between the smallest elements and the largest ele-ments. The number of elements, Ne, increases from the coarsest mesh of 30,772 to the finest mesh of 607,102 for which the CPU time for the generation of nodal points, the construction of the triangular elements and the graphics display has also increased from 2.23 to 750 s.

Two partitions of background grid 30 × 20 and 60 × 40 have been tested, and it is found that, in general, the CPU time could be reduced by more than four times depending on the complexity of the mesh, as shown in the column of efficiency in Table 3.1. As expected, the major gain in the CPU time is attributed to searching on the boundary segments for intersections and locating a candidate node for the formation of a new element. Referring to Table 3.1, by means of the background grid, Nc is usually a small fraction of Nb, which accounts for the drastic reduction in the searching time. However, the difference between the two grids 30 × 20 and 60 × 40 is pretty small, showing that efficiency is not sensitive to grid resolution as long as it is sufficiently fine and reasonable.

For the two grids, the segments retrieved in the intersecting cells are more or less the same, indicating that potential line segments could be accurately determined by a relatively coarse grid, even though the number of cells intersected by the base line segment might not be the same for the two different grid settings. Meshes generated with and without a background grid are not identical as the sequence of line segments, and hence, the sequence of element construction might be altered in the presence of a background grid or using grids of different resolutions.

The number of elements and nodes and the resulting mean element shape factor α are, however, more or less the same for meshes generated with and without a background grid, as shown in Table 3.1, indicating that mesh characteristics are not affected by the use of a background grid.

Mean spacing conformity coefficients of line segments for the meshes are also given in Tables 3.1 and 3.2, which almost did not change whether a background grid has been used or not in MG. This implies that the node spacing conformity only depends on how elements are created but not on how the validity of the elements is assured in the formation process.

For the example meshes that have been generated in this study, the mean value of the node spacing conformity of the line segments in the meshes is approximately 12% short of the ideal situation, whereas the mean element shape factor for the mesh is fairly high at about 0.9. For the given domain and the nodal spacing functions tested, there is no analytical theo-retical value for optimal node spacing conformity, and the values quoted could be taken only for reference; a direct comparison with other meshes might not be appropriate.

Figure 3.51 FE mesh for the second nodal spacing function, N_n = 117,496, N_e = 234,940.

Although the pattern of nodal density distribution is more complicated than that of the first node spacing function, trends and observations for the second node spacing function are very much similar to those that have been found for the first series of meshes. The num-ber of elements, N_e, varies from the coarsest mesh of 45,132 to the finest mesh of 944,832, for which the CPU time has also increased from 4.39 to 1945 s, as shown in Table 3.2. Two background grids 30 × 20 and 60 × 40 have also been tested with which the MG time could be reduced by four to six times depending on the mesh characteristics. Again, the reduction in the CPU time is mainly ascribed to the reduced search on boundary segments in the ele-ment construction. As shown in Table 3.2, N_c is only a tiny fraction of N_b, which accounts for the substantial reduction in the CPU time for the MG. For the second series of meshes based on a different node spacing distribution, the difference between the two grids 30 × 20 and 60 × 40 is again very minor, and both grids are equally apt for the purpose of improving efficiency in ADF MG.

High-quality gradation meshes could be generated for rather complicated node spacing functions, as shown in Figures 3.52 and 3.53. The results for MG with and without the use

Figure 3.53 Mesh 4: Nn = 155,783, Ne = 311,304.

Figure 3.52 Mesh 3: N_n = 66,379, N_e = 132,428.

of a background grid are listed in Table 3.2. The gains in CPU time are 3.7 times and 5.3 times, respectively, for these two meshes, showing that the efficiency of the background grid reduces for more complicated node spacing functions whose computation may take a substantial portion of the overall MG time.

Upon further analysis on the use of the two background grids, it is found that identical sets of potential segments susceptible to intersections adjacent to the base segment could be accurately retrieved by both grids, verifying the reliability of the procedure in marking and unmarking of cells intersected by line segments. Similar to the first series of meshes, the use of background grid has no bearing on the characteristics of the resulting meshes, such that the number of elements N_e, the number of nodes N_n and the mean shape factor α are almost exactly the same.

As a practical example of adaptive refinement FE analysis, the electromagnetic wave dis-tribution over a planar domain is considered. A uniform plane wave is scattered by a circu-lar cylinder and propagates freely towards infinity, as shown in Figure 3.54. The analytical solution expressed in cylindrical co-ordinates to this problem was given by Balanis (1989).

E E j J k J k r

Linear triangular elements T3 were used in the adaptive refinement analysis, and the required accuracy was set equal to 1%. Four analyses in three steps of refinements were required to bring the energy error norm from 22.7% to about 1%, as shown in Table 3.3.

The refinement adaptive triangular meshes generated by the proposed dynamic grid tech-nique are shown in Figure 3.55. It is seen that there is little difference between the exact error norm and the estimate error norm (η η/ ), showing that the meshes are of very high quality, and the required nodal spacing over the domain is well respected.

A – A

Figure 3.54 Wave direction and the cylindrical domain and its cross section.

Table 3.3 Adaptive refinement analysis of the electromagnetic wave problem

Mesh DOF NN NE ||e||* η(%) ||e|| η(%) η η/

1 308 324 576 0.089047 24.10 0.084717 22.69 1.06 2 3695 3755 7338 0.013265 3.43 0.011950 3.09 1.11 3 7069 7135 14,092 0.009856 2.54 0.009569 2.47 1.03 4 13,872 13,985 27,350 0.004561 1.17 0.004472 1.15 1.02 Note: DOF = number of degrees of freedom; NN = number of nodes; NE = number of elements;

||e||* = estimated energy norm; ||e|| = exact energy norm; η = estimated % error; η = exact % error.

In the second example of adaptive refinement analysis, two co-planar but non-overlapping wedges are considered. As shown in Figure 3.56, the geometry is composed of two cylin-drical wedges, which are prismatic along the z-direction. The problem was analysed in a domain of radius R = 1.5λ using quadratic triangular elements T6. For adaptive analysis, the target accuracy of the FE solution is set at 2%. The relative error of the initial mesh

Incident wave

y

(x₁, y₁)

(x₂, y₂)

α1 = 30˚

α₂ = 20˚

z x

H ⁱ

E ⁱ

Figure 3.56 Co­planar wedges.

(a) (b)

(c) (d)

Figure 3.55 Adaptive refinement meshes of hollow cylinder: (a) mesh 0, NN = 324, NE = 576; (b) mesh 1, NN = 3755, NE = 7338; (c) mesh 2, NN = 7135, NE = 14092; (d) mesh 3, NN = 13985, NE = 27350.

Table 3.4 Adaptive refinement analysis of co­planar wedges

Mesh DOF NN NE η(%) η(%) η η/

1 1203 1275 594 13.24 16.50 0.80

2 4487 4621 2228 6.93 8.81 0.79

3 13,153 13,376 6530 2.03 2.33 0.87 4 17,107 17,456 8563 1.65 1.72 0.96

was about 16.5%, and the meshes achieved the required accuracy in three refinements, as shown in Table 3.4. The meshes generated by the AFT in the adaptive procedure are given in Figure 3.57.

3.6.5 Closure

A dynamic process has been developed such that the number of objects recorded in each cell of the background grid is a variable depending on the actual need as required by the node spacing function. The cells have to be marked and unmarked whenever there is a change in the generation front upon the formation of a new element; thus, the line seg-ments contained in each cell change continuously throughout the entire process of MG.

This scheme drastically reduces the memory requirement for the implementation of a background grid of a higher resolution of cell partitions. From meshes of various char-acteristics, it is found that an integer array of size equal to the number of elements to be generated is all that is required for the background grid construction, which is quite worthwhile for the drastic reduction in CPU time for MG by more than five times, which could have been even higher if the nodal point generation time were excluded in the mesh-construction time.

The dynamic background grid scheme could be easily added on to any existing MG programs based on AFT to boost efficiency, as the main stream flow of the MG pro-cedure would not be affected at all, except that the search for entities could now be localised and be made much more efficient than before with the aid of a dynamic grid.

Obviously, the idea could be further extended to the case of MG in 3D and over curved surfaces following exactly what is done in 2D. The only modification is that the domain of interest has to be partitioned into 3D cells whose number is given by N = N_x × N_y × N_z, where N_x, N_y and N_z are, respectively, the subdivisions along the three principal directions.

Figure 3.57 Adaptive refinement finite element meshes for co­planar wedges.

3.7 MESHING BY A COMBINED SCHEME